D-Spectrum and D-Energy of Complements of Iterated Line Graphs of Regular Graphs

The D-eigenvalues {µ۱,…,µp} of a graph G are the eigenvalues of its distance matrix D and form its D-spectrum. The D-energy, ED(G) of G is given by ED (G) =∑i=۱p |µi|. Two non cospectral graphs with respect to D are said to be D-equi energetic if they have the same D-energy. In this paper we show that if G is an r-regular graph on p vertices with ۲r ≤ p - ۱, then the complements of iterated line graphs of G are of diameter ۲ and that ED(\overline{Lk(G)}), k≥۲ depends only on p and r. This result leads to the construction of regular D-equi energetic pair of graphs.

Keywords:

Distance spectrum , Distance energy , Line graphs